The quadratic fields with discriminant divisible by exactly two primes and with ``narrow'' class number divisible by 8

Abstract

Let $K$ be the quadratic field $\mathbf{Q}(\sqrt{m})$ with discirimant $d = pq$. Using Legendre's theorem on the solvability of the equation $ax^2 + by^2 = z^2$, we give necessary and sufficient conditions for the class number of $K$ in the narrow sense to be divisible by 8. The approach recovers known criteria but is simpler and can be extended to compute the sylow 2-subgroup of the ideal class group of quadratic fields.